How to Set Shape or Direction of Particle Trail
In your particular case (with a spherical emitter) you might be able to get away with simply reversing the sign of the initial particle Velocity along the Normal. With a positive Normal velocity the particles will spread out from the surface. However, a negative Normal velocity will start the particles in a direction towards the centre of your mesh so they will naturally converge on each other. You will need to tweak the strength, particle life span and/or damping so that they don't cross over.

EDIT : To retain the 'engulfing' effect of the emitted particles, you can emit the particles from a separate (non-rendered) mesh that surrounds the original object. This will allow the particles to be emitted a distance away from the surface so that they surround it rather than emanating from within.

Note that the emitter mesh is set to not render.

To fill in the gap between the surface and where the trail particles are emitted you can add a second particle system - emitting short lived particles from the surface to the point at which the trail particles are emitted. This will make the object appear to &quot;fizz&quot; with particles and will mask the fact that the trail particles are actually emitted at a distance from the surface. Increase the number of particles to get the desired density.

This can produce the following result :

You could add Brownian motion or a turbulance field or other fields so as to make the particle path less linear.
Blend file attached.